Uniform spanning forests associated with biased random walks on Euclidean lattices

The uniform spanning forest measure ($\mathsf{U}\mathsf{S}\mathsf{F}$) on a locally finite, infinite connected graph $G$ with conductance $c$ is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding $\mathsf{U}\mathsf{S}\mathsf{F}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that on ${\mathbb{Z}}^d$, equipped with the unit conductance $c=1$, $\mathsf{U}\mathsf{S}\mathsf{F}$ is concentrated on spanning trees if and only if $d\le 4$. In this work we study the $\mathsf{U}\mathsf{S}\mathsf{F}$ associated with conductances induced by $\lambda$–biased random walk on ${\mathbb{Z}}^d$, $d\ge 2$, $0<\lambda <1$, i.e. conductances are set to be $c(e)={\lambda }^{-\mathrm{\mid }e\mathrm{\mid }}$, where $\mathrm{\mid }e\mathrm{\mid }$ is the graph distance of the edge $e$ from the origin. Our main result states that in this case $\mathsf{U}\mathsf{S}\mathsf{F}$ consists of finitely many trees if and only if $d=2$ or $3$. More precisely, we prove that the uniform spanning forest has $2^d$ trees if $d = 2$or$3$,andinfinitelymanytreesif$d \geq 4$. Our method relies on the analysis of the spectral radius and the speed of the $\lambda$–biased random walk on ${\mathbb{Z}}^d$.